Downhole microseismic monitoring of hydraulic fracturing: a
full-waveform approach for complete moment tensor inversion and stress
estimation
F. Song, M. N. Toksöz & J. Li
Earth Resources Laboratory, MIT, Cambridge, USA
ABSTRACT: Downhole microseismics has gained in popularity in recent years as a way to characterize hy-draulic fracturing sources and to estimate in-situ stress state. Conventional approaches only utilize part of the information contained in the microseismic waveforms such as the P/S amplitude ratio and/or P first motion polarity to determine the microearthquake focal mechanisms and infer stress state. Thus, additional con-straints like double-couple assumption must be made to stabilize the inversion for conventional methods. The situation becomes even worse for downhole monitoring where only limited azimuthal coverage is available. In this study, we have developed a full-waveform based approach to invert for complete moment tensor. We use the discrete wavenumber integration approach as the fast forward modeling tool to calculate the synthetic waveforms for one-dimensional layered velocity models. By matching full three-component waveforms across the array, a stable moment tensor solution can be obtained without imposing additional constraints. We also derive the source radius from the far-field displacement spectrum with the Madariaga’s model and de-termine the stress drop afterwards. We test our method on a downhole microseismic dataset from hydraulic fracturing treatments in East Texas. The result indicates the existence of the isotropic component in some events. A clear difference is observed that non-double-couple events tend to have smaller stress drops, which is consistent with other studies. The derived fracture plane direction also agrees with that derived from mul-tiple event location.
1 INTRODUCTION
Microseismic downhole monitoring is a valuable tool for mapping the fractures and evaluating the ef-fectiveness of hydraulic fracturing. The locations of microseismic events, with sufficient resolution, pro-vide information on fracture geometry and proper-ties (Warpinski et al. 1998, Phillips et al. 2002).
However, additional information besides location is contained in the microseismic waveforms. For ex-ample, seismic moment tensor has gained more and more interests recently in terms of understanding the microseismic source mechanisms and stress state (Nolen-Hoeksema & Ruff 2001, Baig & Urbancic 2010). Although moment tensor inversion has been applied in downhole hydraulic fracturing monitor-ing, most of them rely only on P- and S- wave am-plitudes and/or P-wave first motion polarities. Due to the limited usage of waveform information, these approaches normally either require multiple moni-toring wells from different azimuths or make double-couple source assumptions in order to stabil-ize the inversion (Vavrycuk 2007, Baig & Urbancic 2010). In most cases of hydraulic fracturing, data are available from only one monitoring well. Given this limitation, the following questions arise: 1) can we
invert complete moment tensor with data from one single well? 2) how can we stabilize the inversion without making the assumption of double-couple sources?
In this paper, we propose a full-waveform ap-proach for moment tensor inversion with one single monitoring well. It uses the discrete wavenumber in-tegration method to calculate the full elastic wave-field in the layered medium. By matching full waveforms across the geophone array, we show that the moment tensor inversion can be stabilized so that the complete moment tensor solution can be re-trieved from an array of three-component geophones in a single borehole. In this paper, we begin by in-troducing the full-waveform based moment tensor inversion approach and testing the method with syn-thetic data to extract seismic moment, fracture orien-tation, and the isotropic component part of the mo-ment tensor. Then we describe the application to a field dataset from East Texas. We invert the full seismic moment tensor and derive stress drop from the far-field displacement spectrum based on Mada-riaga’s model (Madariaga 1976).
2 METHODOLOGY
2.1 Full waveform based moment tensor inversion The moment tensor of microseismic events can be represented by a 3 by 3 symmetric matrix mjk (Aki
& Richards 2002). To improve the inversion with a single borehole coverage, we use all phases that are embedded in the full waveform data. Our approach starts from full elastic waveform modeling in the layered medium with discrete wavenumber integra-tion method (DWN; Bouchon 2003). The i-th com-ponent (North, East, Down) of the observed wave-form at geophone n is modeled as:
( )
t s j t s x n r x k ij G k jk m t s x n r x i v ∑ ∑ = = ⊗ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 3 1 , , , 3 1 , , (1) where ⎟ ⎠ ⎝xr xs k ijG , , , is the i-th component of the
Green’s function at geophone from a point mo-ment tensor source mjk at
⎞ ⎜⎛ n t n r x s
x , s
( )
t is the source timefunction. In this study, a smooth ramp function is used as s
( )
t .The misfit function for inverting moment tensor matrix mjk is defined by:
dt N n i n T t s x n r x i v t s x n r x i d jk m ∑ ∑ ∫ = = ⎟⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Δ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 1 3 1 2 0 , , , , ) ( (2) where ⎟ ⎠ s x
r is the observed data, while
⎟ ⎠ ⎜
⎝xr xs i
v , , is the synthetic data as described in
Eq-uation 1. Tn is the duration of observed waveforms
at geophone n. In this study we choose Tn to include
both P and S wave trains and is constant for all geo-phones. Time 0 is the origin time, which is obtained by grid-search around its initial estimate within the dominant signal period. The initial estimate of the origin time can be found by cross-correlating the synthetic and observed waveforms. To further stabil-ize the inversion, we band-pass filter both synthetic data and observed data to the signal frequency band, which we select [200, 900] Hz in this study. The moment tensor is solved by minimizing the misfit function in Equation 2 as:
⎞ ⎜ ⎝ ⎛ t n x i d , , ⎞ t ⎛ n ) ( ) ( 1 ) ( s x j D s x ij A s x i M ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = (3)
Here Mi is the i-th component of six independent
moment tensor elements: M1 = m11, M2 = m22, M3
= m33, M4 = m12, M5 = m13, M6 = m23, while Dj has
six independent elements:
dt k d t s x n r x N n k n T kj g s x j D x xs t n r ) ( , , 1 3 1 0 ) ( ⎟ , , ⎠ ⎞ ⎜ ⎝ ⎛ ∑ ∑ ∫ = = = (4)
where j = 1, 2, 3…6 and gkj corresponds to one of
the six elementary seismograms and each is defined by:
( )
, 1,2,3 ) ( , ) ( , , = , , ⊗st j= j kj G kj g x x t x xs t n r s n r (5)( )
t s k G k G k g x xs t n r )=[ 1,2 + 2,1]⊗ ( 4 , , (6)( )
t s k G k G k g x xs t n r )=[ 1,3 + 3,1]⊗ ( 5 , , (7)( )
t s k G k G k g x xs t n r )=[ 2,3 + 3,2]⊗ ( 6 , , (8)Finally A is a 6*6 matrix with elements:
dt ki g t s x n r x N n k n T kj g ij A x x xs t n r s , , ( ) 1 3 1 0 ) ( ⎟ , , ⎠ ⎞ ⎜ ⎝ ⎛ ∑ ∑ ∫ = = = (9)
Once we obtain the full moment tensor matrix , we can obtain seismic moment as:
jk m i M0 =maxλ (10) where i
λ is the eigenvalue of moment tensor matrix. The moment magnitude is defined as:
067 . 6 ) 0 ( 10 log 3 2 − = M w M (11)
where M0 is the seismic moment in N*m. The full
moment tensor matrix is further decomposed into three parts: isotropic (ISO) component, compensated linear vector dipole (CLVD) component and double-couple (DC) component (Vavrycuk 2001). The frac-ture strike is then derived from the DC component (Jost & Hermann 1989). The ISO percentage is ob-tained by: % 100 0 ) ( 3 1 M jk m tr ISO c = (12)
2.2 Stress drop estimation
According to (Madariaga 1976), the radius of a cir-cular source can be estimated from:
c f s V r π 2 32 . 1 0= (13)
where fc is the corner frequency derived from
S-wave displacement spectrum. Stress drop, defined as the average difference between the initial and final stress levels over the fault plane, is obtained from:
3 0 0 16 7 r M = Δσ (14)
where M0 is the seismic moment determined by full
0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8 Time (s) G e op ho ne i nd ex Synthetic data Inverted data a) P S P/S,S/P Converted Wave 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8 Time: s G eophon e i ndex Synthetic data Inverted data b) P S P/S,S/P Converted Wave 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8 Time (s) G eo phone i nde x 3 SYNTHETIC STUDY
3.1 Full waveform fitting and seismic moment
tensor inversion
Synthetic data Inverted data P S P/S,S/P Converted Wave c)
For the synthetic study, we use DWN to generate clean data. We add 10% Gaussian noise to form the synthetic data. Figure 1 shows the 1D velocity model used for the synthetic study, which is identic-al to the following field study.
The red curve in Figure 2 shows the synthetic three-component data for a double-couple source (strike: 85 degrees, dip: 75 degrees, rake: 0 degree) at a vertical depth of 3975.3 m and 64.8 m north, 77.3 m west away from the six-geophone array in the vertical monitoring well. The inverted data is shown in blue for each plot. We see a pretty good agreement between synthetic data and inverted data. Besides fitting the P- and S-wave, we also see a good fit for P & S converted wave. This additional waveform information helps constrain the inversion and lower the condition number of the matrix A. This is also the reason why we can invert for full moment tensor from one vertical geophone array without making additional source assumptions. The inverted moment tensor gives a fracture plane with strike of 84.9 degrees, dip of 74.5 degrees, and rake of -0.6 degrees, which is pretty close to the true val-ue especially the strike valval-ue. This shows the validi-ty of our approach in extracting fracture geometry information, among which strike information is the most interesting. 2000 3000 4000 5000 6000 3800 3850 3900 3950 4000 4050 4100 Velocity: m/s De pt h: m S wave velocity P wave velocity
Figure 1. One-dimensional P- and S-wave velocity model for both synthetic study and field study, which is derived from
ell logging data. w
e curve gives the fitted data from moment tensor inver-on.
Figure 2. Three-component velocity waveform fitting: a) North component, b) East component, c) Down component. The red curve shows the synthetic data with 10% Gaussian noise, while
e blu th si
imator. For ach seismic moment value, a full-waveform mo-3.2 Seismic moment estimate
To evaluate the accuracy of the seismic moment es-timated by our method, we conduct the Monte-Carlo simulation. In this experiment, we fix the source lo-cation at (N,E,D) = (64.8, 77.3, 3975.3) m and a source mechanism of (strike, dip, rake) = (85, 75, 0) degrees. We vary the seismic moment from 103 N·m to 108 N·m. For different seismic moment values, different realizations of 10% Gaussian noise are added to the synthetic data to study the statistical
roperties of the seismic moment est p
e
ment tensor inversion as described above is per-formed to retrieve the seismic moment.
103 104 105 106 107 108 103 104 105 106 107 108
True seismic moment (N*m)
E s ti ma te d s e is mic mo me n t ( N *m )
eigenvalue decomposition as de-scribed in Equation 10. In summary, for a broad range of seismic moment values, our method gives a good estimate.
Figure 3. Comparison of true seismic moment and estimated seismic moment.
Figure 3 gives the comparison between true seis-mic moment used to generate synthetic data and the estimated seismic moment by full moment tensor in-version. The estimated seismic moment values agree well with the true seismic moment values. The mean error is around 3.9%, while the standard deviation is about 5.2%. This means that the true values of M0
are within the confidence region of the estimated M0 under additive Gaussian noise, which is
consis-tent with previous study (Patton & Aki 1979). The non-zero bias comes from two parts: 1) the errors propagated into the moment tensor inversion solu-tion due to the additive data noise, which is well bounded by the illness of matrix A; 2) the numerical errors from the
3.3 Strike estimate
A similar Monte-Carlo simulation has been con-ducted to assess the performance of strike estimator. For this test, we test two source mechanisms. One is the pure double-couple mechanism while the other is a 60% double-couple component with a 40% iso-tropic component. For both scenarios, the dip and rake value is set to be 75 and 0 degrees. We also fix the source location same as in Section 3.2 and use a constant seismic moment of 5*104 N·m. The strike
value is changed from 0 to 360 degrees. For each strike value, a forward modeling with 10% additive Gaussian noise is conducted and a full-waveform moment tensor inversion is performed afterwards to invert for the full moment tensor and retrieve the
trike. s 50 100 150 200 250 300 350 50 100 150 200 250 300 350
True strike (Degrees)
E s ti m a te d s tri k e (De g re e s )
Figure 4. Comparison of true strike and estimated strike (pure C source). D 50 100 150 200 250 300 350 50 100 150 200 250 300 350
True strike (Degrees)
E s ti m at ed s tr ik e ( D egr ees )
Figure 5. Comparison of true strike and estimated strike (source: 60% DC component + 40% ISO component).
The estimated strike value is plotted against the the true strike value for both cases in Figures 4 and
5. The estimated strike is in good agreement with the true strike. The estimated strike has a mean error of -0.1 degrees from the true strike for the pure DC source, while the mean error increases to 0.3 degrees for the isotropic plus double-couple source. The in-creased error in strike is due to the dein-creased devia-toric part of the seismic moment tensor, from which the strike is defined.
3.4 Isotropic component percentage estimate The isotropic component percentage defined in Equ-ation 12 provides a good indicator of fracture volu-metric strain. It gives some idea about fracture open-ing and closure. As discussed in Section 3.3, it also indicates the uncertainty of strike estimates to some degree. The higher the isotropic component percen-tage is, the less accurate the strike estimate is.
In this simulation, we change the relative percen-tage between DC component and the ISO compo-nent and fix the seismic moment to be 5*104 N·m. All other parameters are kept the same as Section 3.2. Figure 6 represents the comparison between true isotropic percentage and the estimated isotropic per-centage. The estimated ISO percentage generally agrees well with the true ISO percentage. There is some scattering for the estimates due to 10% Gaus-sian noise. The mean error is around -0.2%, while the standard deviation approaches 2.6%. This means the true isotropic percentage value falls into the con-fidence region of the estimated isotropic percentage,
lthough the maximum absolute error is close to 8%. a 0 10 20 30 40 50 60 70 80 90 100 -20 0 20 40 60 80 100
True isotropic percentage (%)
Es ti m a te d i s o tr o p ic p e rc e n ta g e ( % )
Figure 6. Comparison of true isotropic percentage and
esti-ma d isotropic percentage. te
4 FIELD STUDY 4.1 Field setup
A microseismic survey was conducted during the hydraulic fracturing treatment of the Bonner sands in the Bossier play at a depth approximately from 3956 m (12980 ft) to 3981 m (13060 ft). The micro-seismic data was collected using a twelve-level, three-component geophone array deployed in the vertical monitoring well at a depth from 3874 m (12710 ft) to 3944 m (12940 ft). The treatment well is approximately 151 m (495 ft) away from the mon-itoring well. The recorded data was analyzed and lo-cated for hydraulic fracturing mapping as outlined by Griffin et al. (2003), and Sharma et al. (2004). The velocity model used for location is shown in Figure 1. In this study, we test our method on sever-al located microseismic events to invert for the com-plete moment tensor and estimate the stress drop from the full waveforms. The microseismic data from the lower six geophones at a depth from 3912 m (12835 ft) to 3944 m (12940 ft) are selected due to their higher signal-to-noise ratios (SNRs). Figure 7 shows the horizontal plane view of the monitoring well at the origin and seven selected microseismic events, which shows a fracture plane mostly along W-E direction. This has been demonstrated in the
aper by Sharma et al. (2004). p -800 -60 -40 -20 0 10 20 30 40 50 60 70 80 S o ut h-N o rt h ( m ) West-East (m)
Figure 7. Horizontal plane view of selected microseismic vents.
e
In next section, we will show the results from full waveform moment tensor inversion and stress drop estimation. We will use one event, named event 1, to demonstrate our procedure. Then we will present and discuss the results for all seven chosen events.
4.2 Seismic moment tensor inversion and stress
drop estimation
As described in section 2, we firstly invert for com-plete moment tensor from full waveforms and ex-tract three important parameters: 1) seismic moment, 2) strike, and 3) isotropic component percentage. Figures 8 a) and b) give the waveform fitting of event 1 between synthetic data and observed data for the north and east component separately. Here only two horizontal components are used in the inversion, because the vertical component has a much poorer SNR due to the poor clamping to the formation. A good agreement of dominant P and S wave trains is seen in both Figures 8 a) and b). The un-modeled wave packages are probably due to the scattering
rom un-modeled lateral heterogeneity. f 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 Time (s) G eo phon e i nd ex Observed Synthetic a) 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 Time (s) G eop hone i nde x Observed Synthetic b)
Figure 8. Comparison between synthetic data and observed da-ta for event 1: a) North component, b) East component.
The estimated seismic moment, strike and the iso-tropic percentage for event 1 are listed in Table 1.
We see a negative isotropic component for event 1, which implies a contribution from implosion. This possibly corresponds to some degree of fracture clo-sure. The seismic moment for event 1 is around 6.1*104 N·m, suggesting a moment magnitude around -2.87. The strike estimated from the devia-toric component gives the values for the fracture plane and the auxiliary plane. It is hard to tell the fracture plane from one single event. The strike val-ues determined from moment tensor inversion for all seven events are listed in Table 1. Compared to the fracture orientation reported in the paper by Sharma et al. (2004), the first set of strike values agrees well with their result of N870E derived from multiple event location and gives the fracture plane strike. The scattering around N870E possibly comes from three factors: 1) velocity model inaccuracy, 2) loca-tion errors, and 3) noise contaminaloca-tion as shown in
ection 3.3. . u _ _ _ e e t orth) _ _ _ _ _ ___ 4.4 -2.96 82 203 ____________________________________________ _ _
inematic model defined by the S
T b Res lts of source parameter determinations.
_ __ ___a le 1 _ _ ___________________________________ Event M0 Mw Strike ____________ __________________ 104 N·m D gr es (Eas of N __ _ ___ _ _ _________________ ____ ___ __ ____ 1* 6.1 -2.87 102 12 2* 2.1 -3.17 80 207 3* 8.1 -2.79 73 196 4* 45 -2.29 139 39 5 5.3 -2.91 75 197 6 7.5 -2.81 95 211 7_ T b 1 continued. ______________________________________________ a le
Event Mw fc r0 ∆σ ISO percentage
_____________________ _____________ Hz m Kpa % ____ _ _ _ _ __ ___ ___ _ ______ _______________ _ _ _____ 1* -2.87 481 1.2 14.9 -26 2* -3.17 561 1.0 8.3 29 3* -2.79 547 1.1 29.2 31 4* -2.29 564 1.0 178.0 30 5 -2.91 714 0.8 42.7 11 6 -2.81 736 0.8 65.8 -10 7_ ___________________________________________ -2.96 744 0.8 39.6 -4
To estimate the stress drop, Madariaga’s (1976) model is adopted to estimate the source radius from the S-wave corner frequency. The recorded voltage data is converted to displacement considering the geophone response (Warpinski 2009). The spectral analysis is then applied to the converted S-wave dis-placement data. For event 1, Figure 9 shows the S-wave displacement spectrum and the best-fit curve determined from the k
following equation: 2 ) ( 1 c f f + 0 ) ( s Q s V fR e f U − ⋅ Ω = π (15)
where R is the source-receiver distance, Qs = 100 is the S-wave quality factor. In the present case of event 1, average values of 2605 m/s and 112 m are accepted for the S-wave velocity and the source-receiver distance separately. A simple nonlinear-least square inversion is deployed to estimate corner frequency fc (Talebi & Boone 1998). Source radius
r0 is then derived from fc according to Equation 13.
The stress drop is finally determined from previous-ly obtained source radius and seismic moment by Equation 14. The stress drop values for all seven events are listed in Table 1. A clear difference is ob-served between estimates of stress release parame-ters for the two types of events as those events hav-ing a significant isotropic component percentage (marked as * in Table 1) tend to have smaller stress drops compared to deviatoric type events of a simi-lar magnitude range. This is consistent with previous studies of injection-induced microseismicity in oil fields (Talebi & Boone 1998). The possible
explana-ion is that the source
t areas associated with events
aving significant isotropic components are more h
likely to have weaker shear strengths, and thus smaller stress drops.
101 102 103 104 10-16 10-15 10-14 10-13 10-12 10-11 Frequency: Hz D is p la c e m e nt s pe c tr al de ns it y : m *s
Figure 9. S-wave displacement spectrum: observed (blue) and
ull-waveform approach has a great otential to improve the source properties study in e situations where only a single monitoring well is
for eir helpful suggestions. We thank Halliburton nergy Services Company and Anadarko Petroleum
ermission to publish this work.
Ba tensors:
Bo Gri
arpinski, N.R. 2003.
Hy-nce and Exhibition, Denver, 5-8 October
Jost
of moment tensors. Seismological Research Letters Ma
Pat
the Royal Astronomical Society 59(3): 479-495.
model fitted (red).
5 CONCLUSIONS
In this paper, we developed a full-waveform based moment tensor inversion approach for hydraulic fracturing monitoring using one single monitoring well. By exploring full waveform information in a one-dimensional layered medium instead of using only P/S amplitude ratio and/or P first motion po-larity, we have demonstrated that the complete mo-ment tensor inversion can be stabilized without mak-ing additional double-couple source assumptions. By synthetic and field test, we have shown that the
strike, isotropic component and stress drop can be reliably derived from this full waveform analysis approach assuming a certain source model. Synthet-ic tests also indSynthet-icate that additive Gaussian noises do not pose difficulties for recovering reliable estimates of the moment tensor. Field data examples show the existence of both deviatoric type events and isotrop-ic type events. In both cases, the derived strike val-ues are in good agreement with the fracture azimuth determined from multiple event locations. However, stress drop studies indicate that isotropic type micro-seismic events tend to have smaller stress drops compared to deviatoric type events of a similar magnitude range. Errors in source parameter esti-mates may come from the inaccuracies in source lo-cations and velocity models. Future work includes further refinement over source locations and veloci-ty models. This f
p th
available.
Acknowledgements
The authors would like to thank Pinnacle - A Halli-burton Service for providing the data and for fund-ing this research. We are grateful to Dr. Norm War-pinski, Dr. Jing Du, Dr. Erkan Ay and Dr. Qinggang Ma from Halliburton Energy Services Company, Dr. Michael Fehler and Dr. William Rodi from MIT th
E
Corporation for p
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